This widely popular textbook provides a mathematically rigorous introduction to analysis of real­valued functions of one variable. This intuitive, student-friendly text is written in a manner that will help to ease the transition from primarily computational to primarily theoretical maths.

Preface. Introduction. 1. The Real Number System. 1.1. Sets and Functions. 1.2. Properties of the Real Numbers as an Ordered Field. 1.3. The Completeness Axiom. 2. Sequences of Real Numbers. 2.1. Sequences of Real Numbers. 2.2. Subsequences. 2.3. The Bolzano-Weierstrass Theorem. 3. Topology of the Real Numbers. 3.1. Topology of the Real Numbers. 4. Continuous Functions. 4.1. Limits and continuity. 4.2. Monotone and Inverse Functions. 5. Differentiation. 5.1. The Derivative of a Function. 5.2. Some Mean Value Theorems. 6. Integration. 6.1. The Riemann Integral. 6.2. Some properties and applications of the Riemann Integral. 6.3. The Riemann-Stieltjes Integral. 7. Series of Real Numbers. 7.1. Tests for Convergence of Series. 7.2. Operations Involving Series. 8. Sequences and Series of Functions. 8.1. Sequences of functions. 8.2. Series of Functions. 8.3. Taylor Series. 8.4. The Cantor Set and Cantor Function. 9. Fourier Series. 9.1. Fourier Coefficients. 9.2. Representation by Fourier Series. Bibliography. Hints and Answers for Selected Exercises. Index.